Vertices-dependent centroid construction
Subsurf Modifier

You can surely set up your own rigging system, but if you don't need to extremely custumize your animation, fastest way to show the position of the centroid is taking advantage of the Subdivison Surface Modifier with Simple type enabled in oder to see the medians meeting in the middle.



Drivers

In alternative, I would suggest to Hook the vertices of a Poly Curve (the triangle) to three Empties and use Drivers to control the location of a fourth object (the centroid). This way you should be able to customize almost everything in your scene (the triangle section, the presence of mendian lines, the shape of the centroid individuator..)



For each coordinate you should assing a value coming from a scripted expression capable of giving the Average mean of the other empties coordinate relative to the considered axis.